International Journal of Assessment Tools in Education
2019, Vol. 6, No. 4, 617–635
https://dx.doi.org/10.21449/ijate.593636
Published at http://www.ijate.net http://dergipark.org.tr Research Article
617
Development of a Preschool Teachers’ Pedagogical Content Knowledge Scale
regarding Mathematics
Hatice Dağlı 1,*, H. Elif Dağlıoğlu 1, E. Hasan Atalmış 1
1Kahramanmaraş Sütçü İmam University, Department of Preschool Education, Turkey 2Gazi University, Department of Preschool Education, Turkey
³Kahramanmaraş Sütçü İmam University, Department of Measurement and Evaluation in Education, Turkey
ARTICLE HISTORY
Received: 18 July 2019
Revised: 07 October 2019
Accepted: 05 December 2019
KEYWORDS
Mathematics education,
Preschool teacher,
Pedagogical content
knowledge
Abstract: This study aimed to develop a measurement tool in order to assess
preschool teachers’ pedagogical content knowledge regarding mathematics.
The study was based on 300 preschool teachers working in formal
independent kindergartens and nursery classes of primary/secondary schools
in the Kahramanmaraş Province of Turkey. Among the participants, 150 were
chosen for pre-application and 150 for the main application. The scale consists
of five different case studies and a total of 35 items, including dialogues that
focus on mathematical content and processes reflected in children’s talk
during their play. In calculating the reliability of the scale, Cronbach Alpha
was found to be .95 for the pre-application and .96 for the main application.
For the validity of the scale, exploratory and confirmatory factor analyses
were performed. The exploratory factor analysis results revealed the scale to
be a single-factor structure. When the factor loads of each relevant item were
examined, no item was found to exist with a factor load value of less than .30.
After confirmatory factor analysis was performed, the model fit indices of
CFI, TLI, RMSEA, and SRMR values were found to be .91, .91, .06 and .06,
respectively. These results show the model to be reliable to an acceptable
level. Based on the findings, it could be concluded that the scale is an
instrument that produces valid and reliable measures, and that it can be used
in order to determine the preschool teachers’ pedagogical content knowledge
regarding mathematics.
1. INTRODUCTION
Mathematics and mathematical thinking have been regarded as key skills of our time in terms
of their scientific field. The development of mathematical competencies begins at birth
(Anthony & Walshow, 2009; Çoban, 2002). Mathematics is a field containing important
concepts and skills which are widely used in learning processes and particularly in daily life.
As people interact with their environment in daily life, they encounter various concepts such as
time, space, shapes, and numbers, and therefore interact with mathematics without even
realizing it (Bulut & Tarım, 2006). Understanding mathematics provides children with the
ability to solve problems and to make correct decisions. Mathematics knowledge requires many
CONTACT: Hatice Dağlı [email protected] Kahramanmaraş Sütçü İmam University,
Department of Preschool Education, 46100, Kahramanmaraş, Turkey
ISSN-e: 2148-7456 /© IJATE 2019
https://dx.doi.org/10.21449/ijate.593636http://www.ijate.net/http://dergipark.org.tr/https://orcid.org/0000-0002-0788-0413https://orcid.org/0000-0002-7420-815Xhttps://orcid.org/0000-0001-9610-491X
Int. J. Asst. Tools in Educ., Vol. 6, No. 4, (2019) pp. 617–635
618
skills such as establishing cause-effect relationships, making calculations, calculating time,
money management, and the use of technology (Ontario Ministry of Education [OME], 2014).
Researchers who conduct studies on cognitive development have revealed that the early
development of mathematical skills closely relates to children’s academic achievement in
subsequent years (Anders & Rossbach, 2015; Aunola, Leskinen, Lerkkanen, & Nurmi, 2004;
Clements, Sarama, & DiBiase, 2004; Gersten et al., 2009). Mathematics is a particularly
hierarchical subject, in which mastery of simple concepts and procedures is required in order
to understand more difficult mathematics (Watts, Duncan, Clements, & Sarama, 2017). It is of
significant importance to teach mathematics that children will use and encounter throughout
their lives (OME, 2014). In this regard, the early childhood years are particularly vital as the
starting point for children to encounter formal mathematics education and basic mathematical
concepts; moreover, mathematical skills are also learned within this period. The aim of
mathematics education at the preschool level is to provide children with meaningful
experiences through gameplay, stories, music and physical activities; to create them a sense of
success with appropriate materials in appropriate physical environments, and to support the
development of mathematics skills without creating a negative attitude towards mathematics
(Arnas, 2006; Dağlı & Dağlıoğlu, 2017; Henniger, 1987; Metin, 1994; Mononen, Aunio, &
Koponen, 2014). Thus, the preschool period is considered to the magical years where children’s
love for mathematics is inculcated and nurtured, and for the development of a positive attitude
towards mathematics.
Some experimental studies on mathematical concepts and skills in the preschool period
depicted that mathematical applications performed with children may create positive
differences in their mathematical competencies when they start primary school, and that these
differences last throughout their school life and even beyond (Anders, Grosse, Rossbach, Ebert,
& Weinert, 2013; Sammons et al., 2004). In this context, mathematics literacy and mathematics
skills are important not only for children’s school success, but also in terms of their professional
career throughout adulthood (Anders & Rossbach, 2015; Clements et al., 2004). Mathematics
education presented to children during their preschool period is significant for their ability to
achieve successful mathematical thinking in the following years and in their readiness
preparation for primary schooling (Claesens & Engel, 2013; Dağlıoğlu, Dağlı, & Kılıç, 2013).
When considered in the long term, understanding mathematics is so effective that it can direct
children towards their future work life and career (Ontario Ministry of Education, 2014).
When it comes to the significance of early mathematics education, recent studies have
emphasized that such education should be structured appropriately to the nature of the child;
and that the child should reach the information by doing and experiencing personally, rather
than teachers attempting to transfer knowledge directly to the child (Arnas, 2006; National
Council of Teachers of Mathematics [NCTM], 2000). In other words, it is necessary for children
to encounter the experiences in which they will learn mathematics concepts by doing and
experiencing during their preschool period (Clements & Sarama, 2014; Umay, 2003).
The recent studies have also suggested that activities prepared in accordance with children’s
interests and their motivation can have a significant effect on their future success (Baranek,
1996; Berhenge, 2013; Mokrova, 2012; Tella, 2007). Education given in subjects or areas that
children are interested in has a more lasting effect (Fisher, 2004). In this regard, mathematics
content can and should play a significant role in early childhood education.
Different research on mathematics education during the preschool period has been conducted
in many different countries. In the Turkish Preschool Education Program, which was updated
in 2013, it is emphasized that mathematics education contributes to the cognitive development
of children, that mathematics education within the preschool period can bestow positive
attitudes in children, and that the mathematical inquiry skills of children can be improved
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through mathematics-based activities (Milli Eğitim Bakanlığı [Turkish Ministry of National
Education], 2013). In addition, mathematical activities that establish relationships between
concepts and life skills should be included in preschool education programs and that child-
centered, game-based and multifaceted activities should be planned.
High quality, interesting and accessible mathematics education for the 3-6-year-olds age group
was emphasized through situational assessments undertaken jointly by the National Association
for the Education of Young Children in America and the NCTM (2010). In particular, the
NCTM emphasized that educational programs which are well-planned, comprehensive, suitable
for children’s development, and meet the required language skills within a cultural context will
be more effective (NCTM, 2009, 2013). Accordingly, mathematical understanding, knowledge
and skills need to be gained during the education period starting from preschool. The NCTM
(2009) also set content and process standards; defined the concepts and contents that children
should learn through the content standards, and concept and content knowledge acquisition as
well as using methods information through the process standards. When based on mathematics
education, the NCTM (2000) showed that mathematical activities and mathematical content
such as numbers, operations, geometry and measurement should be integrated with process
standards such as problem solving, reasoning and proof, association, communication and
symbolization. This process showed that mathematics program and education practices should
be structured on a sound basis by taking into account both mathematical content areas and the
developmental characteristics of children.
Considering that pedagogical approaches supporting the development of mathematical skills
are seen as effective in enhancing these skills in children (Mononen & Aunio, 2013); the
importance of teachers’ pedagogical content knowledge related to mathematics has become
prominent (Gifford, 2005). Pedagogical content knowledge in education was originally
proposed by Shulman (1986), and encompasses knowing what to teach according to age groups
and integrating that with the knowledge of how to teach it.
McCray (2008) explained the factors affecting pedagogical content knowledge regarding
mathematics, as illustrated in Figure 1.
Figure 1. Pedagogical Content Knowledge Related to Mathematics (as revised by McCray, 2008)
McCray (2008) defined pedagogical content knowledge regarding mathematics as a junction
point of three questions in mathematics education; Who will teach?, What to teach?, and How
to teach? Teacher’s pedagogical content knowledge, content knowledge and teaching ability
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are of paramount importance in effective learning and success in children (Jang, 2013; Zhang,
2015). The basis of mathematics education begins with an understanding of mathematical
knowledge (Zhang, 2015).
Teachers with pedagogical mathematics concept knowledge know which concepts are the most
basic and the best analogies that can gain help conceptual understanding; can enter events with
new ideas in accordance with the interests of children, and ensure children use mathematics and
mathematical language by asking children the right questions (McCray, 2008). The language
used by teachers in the classroom should be founded upon improving the mathematical thinking
of children. Guidelines for teachers in terms of mathematics contents such as numbers, spatial
relationships, and operations should help children to use mathematics (Clements & Sarama,
2014; McGrath, 2010). Teachers should provide support to enable children to develop a positive
attitude towards mathematics by taking full account of mathematics education and preparing
appropriate programs in this regard (Copley, 2010; Dağlıoğlu, Genç, & Dağlı, 2017).
Previous studies show that teachers’ attitudes, pedagogical field knowledge and beliefs affect
children’s mathematical ability, that the methods and techniques used by teachers affect
children’s ability in this field, and that teachers are lacking in mathematics education and in
recognizing children’s abilities (Chace Pierro, 2015; Cox, 2011; Erdoğan, 2006; Güven, 1998;
Hacısalihoğlu Karadeniz, 2011; Kilday, 2010). From analyzing the relevant literature, a few
studies have been specifically focused on preschool teachers’ pedagogical content knowledge
regarding mathematics (Cox, 2011; Kilday, 2010; McCray, 2008; Platas, 2008). However,
studies that have been conducted in Turkey usually focus on preschool teachers’ attitudes,
beliefs and self-efficacy towards mathematics education (Çelik, 2017; Güven, Karataş, Öztürk,
Arslan, & Gürsoy, 2013; Karakuş, Akman, & Ergene, 2018; Koç, Sak, & Kayri, 2015; Şeker &
Alisinanoğlu, 2015); whereas only two studies considered pedagogical content knowledge
regarding mathematics (Aksu & Kul, 2017; Parpucu & Erdoğan, 2017).
The current research was planned in order to develop a measurement tool for determining
preschool teachers’ pedagogical field knowledge of mathematics in order to address a gap in
this field.
2. METHOD
This section includes information related to the working group, the data collection tool, and the
development process of the scale.
2.1. Working Group
The participants of the study consisted of 300 teachers working in formal independent
kindergartens and nursery classes of primary/secondary schools under the Turkish Ministry of
National Education in Kahramanmaraş Province, Turkey; specifically, the districts of
Dulkadiroğlu and Onikişubat. Of the participant teachers, 150 were selected for the pre-
application and 150 for the main application.
While determining the size of the group to conduct factor analysis in the preliminary
application, the researchers proposed different approaches; some argued that there should be
twice the number of items (Büyüköztürk, Kılıç-Çakmak, Akgün, Karadeniz, & Demirel, 2008),
some four times the number of items (MacCallum, Widaman, Preacher, & Hong, 2001), and
others suggested 10 times (Nunnally, 1978). In addition, for exploratory factor analysis, Kaiser-
Meyer-Olkin Test is expected to be greater than .50 and Bartlett test should be statistically
significant (Büyüköztürk, 2010). In this regard, the decision was made to conduct an application
with 150 teachers as the pre-application stage. Table 1 presents the demographic information
regarding the participants.
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Table 1. Participants’ Demographic Information
Demographic Information Pre-application Main Application
Frequency Percentage Frequency Percentage
Gender Female 130 86.7 150 100.0
Male 20 13.3 - -
Age (years)
Less than 20 40 26.7 14 9.3
21-25 47 31.3 42 28.0
26-30 45 30.0 60 40.0
31-35 18 12.0 33 22.0
36 or over - - 1 0.7
Graduation
High School / Girls’ Vocational High School - - - -
Associate Degree Child Development
Vocational School 23 15.3 8 5.3
Undergraduate Preschool Teacher 81 54.0 99 66.0
Undergraduate Child Development
Teacher
41 27.3 34 22.7
Postgraduate 5 3.4 6 4.0
Other - - 3 2.0
Seniority (years)
Less than 1 year 25 16.7 4 2.7
1-5 years 36 24.0 23 15.3
6-10 years 59 39.3 64 42.7
11-15 years 18 12.0 37 24.6
16-20 years 8 5.3 13 8.7
21 years or more 4 2.7 9 6.0
Institution
Independent kindergarten 106 70.7 90 60.0
Primary school 27 18.0 50 33.3
Secondary school 17 11.3 10 6.7
Number of
Children
(per class)
5-10 2 1.3 5 3.3
11-15 12 8.0 14 9.3
16-20 74 49.3 67 44.7
21 or more 62 41.3 64 42.7
Age of Children
36-53 months 15 10.0 - -
54-60 months 70 46.7 308 51.3
61-66 months 65 43.3 292 48.7
Mathematical
Activities
Never - - - -
One time per 2-3 weeks - - 3 2.0
Twice a week 41 27.3 38 25.3
Three to four times a week 86 57.3 73 48.7
Daily 23 15.4 36 24.0
Table 1 shows that 86.7% (n = 130) of the participant teachers were female, whilst 13.3%
(n = 20) were male for the pre-application stage; whereas, all participants were female for the
main application. In the pre-application, 10% (n = 15) of the teachers were working with
children aged between 36 and 53 months, 46.7% (n = 70) between 54 and 60 months, and
43.3% (n = 65) between 61 and 66 months. In the main application, 51.3% (n = 308) of the
teachers were working with children aged between 54 and 60 months, whilst the other 49.7%
(n = 292) were working with children aged between 61 and 66 months.
2.2. Data Collection Tool
The research data was collected through a “Teacher Information Form” and a developed
“Preschool Teachers’ Pedagogical Content Knowledge Scale regarding Mathematics
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(PTPCKSM)” that included five case studies. The Teacher Information Form was used in order
to record the teachers’ gender, age, type of graduation school, their seniority, type of institution
where they were assigned, the number of children in each class, the age group of the children
in their class, and the availability of a mathematics center in class.
The PTPCKSM was developed in order to identify teachers’ awareness towards mathematical
content and the processes involved in language used by children. In this section, five case
studies were designed based on children’s dialogues including different mathematical contexts
and processes from the expressions used by children during play. A separate marking form was
created for each case study and teachers were requested to mark the mathematical contents and
processes they identified in accordance with the form. Based on the NCTM (2000) standards,
the case studies of the PTPCKSM included “counting, geometry, spatial perception, part–whole
relationships, matching, classification/grouping, comparison, sorting, measurement, operation,
pattern, and graphics” as the mathematics contents, and “communication, association,
reasoning and proof, problem solving and representation/symbolization” as the mathematical
processes. Each case study consisted of seven statements/items.
During the scale’s development process, first the existing literature was reviewed. The contents
and processes involved in mathematics education during the preschool period in Turkey and
elsewhere were examined, and the sub-dimensions for the PTPCKSM were formed after
determining the problem statement based on the aforementioned content. The scale known as
“Knowledge of Mathematical Development” that was developed by Platas (2008) for the
purpose of measuring teachers’ knowledge on the development of mathematical concepts in
children was taken as the basis for the current study. Along with the necessary permissions in
the ongoing process, the “Preschool Mathematics-Pedagogical Concept Information Interview
Form” that was developed by McCray and Chen (2008) was also taken into consideration.
Two of the five case studies in the PTPCKSM were prepared based on the Preschool
Mathematics-Pedagogical Concept Information Interview Form; with the other three case
studies formed by the researcher. The case studies were designed based on a straight line
approach, from simple to complex. Each case study contained different yet simple images in
order to add clarity to the case studies. With examination of the content and process standards
developed by the National Association for the Education of Young Children in America and
NCTM, as well as the involvement in the development process of mathematical concepts in
children, significant attention was paid to the inclusion of these standards in case study. Within
the scale development process, three different scale drafts were prepared in the form and coding
dimension, and each draft was applied to three different preschool teachers. The scale was then
finalized by testing the clarity of the scale with various applications.
The expert opinion of seven specialists in mathematics and preschool education, who were also
faculty members at different universities, were obtained in the preparation of the scale. In
addition, a measurement and evaluation expert plus and two Turkish linguists were employed
to examine the scale in terms of the language clarity and application of the items. The scale was
considered ready for the pre-application stage after having taken a total of nine expert opinions.
In the PTPCKSM, spelling errors and incoherencies were corrected so as to increase the scale’s
clarity along with the expert opinion. In order that the data could be grouped and the correct
comparisons made, International Standard Classification of Education (ISCED) categories
(Türkiye İstatistik Kurumu Başkanlığı [Turkish Statistical Institute], 2012) were employed. In
addition, categories (as mathematics contents and mathematical processes) were created for
some related items.
One of the case studies is presented in Figure 2, and the scoring table in the marking form
created for the teachers to record their answers is shown in Figure 3. In each case study, first
the image and the text were taken as a whole; then, in the marking section, each sentence (item)
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was taken separately by dividing each sentence (item) in the case study. Here, the teachers were
expected to see the whole, then the application was made by dividing the text into sentences in
order to be more easily recognize the details in the text.
Figure 2. PTPCKSM Case Study Form
Figure 3. PTPCKSM Scoring Table
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As can be seen in the case study shown in Figure 2 and Figure 3, the PTPCKSM contains one
or more mathematical contents/processes within each sentence. Each item was rated as 1 point
in the scale; therefore, the whole case study was calculated as a total of 7 points, with each case
study consisting of seven items. The pedagogical content knowledge of the teachers with the
highest total score were therefore expected to be considered as high.
2.3. Data Collection Process
In both the pre-application and the main application, educational institutions were visited by
the researcher where the necessary permissions had been received. The school principals were
informed about the study first, and then interviews were subsequently held with each participant
teacher. The scale was introduced to the teachers in person by the researcher, and any necessary
corrections to the scale were applied together. The teachers requested additional time in order
to better complete the scale. The forms were delivered to the teachers by the researcher, and
later retrieved according to pre-specified dates.
2.4. Data Analysis
IBM’s SPSS 22 statistical package and the Mplus 7.4 program were used in order to calculate
the reliability and validity of the developed scale. Cronbach Alpha coefficient was used to test
the reliability of the scale, and then item difficulty index values and item discrimination
coefficients were calculated separately for each item. In order to calculate the validity of the
scale, the content and construct validity were examined; both exploratory and confirmatory
factor analyses were performed for this purpose. Kaiser-Meyer-Olkin Test and Bartlett Tests
were performed in exploratory factor analysis (EFA); whilst CFI, TLI, RMSEA and SRMR
values were calculated for the scale’s confirmatory factor analysis (CFA).
Before scoring, each mathematics content and process in the scale was alphabetically coded as
a, b, c, d, e, f…….p. Table 2 shows the codes corresponding to the mathematical contents and
processes.
Table 2. PTPCKSM Content/Process Coding Values
Mathematics Content/Process Coding Value
There is No Mathematical Statement and Skill a
Counting b
Geometry c
Spatial Perception d
Part–Whole Relationships e
Matching f
Classification/Grouping g
Comparison h
Measurement i
Operation j
Pattern k
Graphic l
Communication m
Association n
Reasoning and Proof o
Problem Solving ö
Representation/Symbolization p
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In the scoring method, each item was awarded equal points (equal scoring) (Frary, 1989;
Masters, 1988).
Each item is worth 1 point, and in scoring the item total is divided by the number of answers
required for its content/process. For example, the per code value of Item 1, in which the correct
answer was “d and m,” is calculated as 1/2 (0.50); and the per code value of Item 2, in which
the correct answer was “b, d and m,” is calculated as 1/3 (0.33).
Any incorrect answer results in 1 point deducted from the total score. For example, a response
of “b, d and n” for Item 3 includes one correct and two incorrect answers; therefore, the score
corresponds to 2 - 1 correct answer is calculated as 1/3 (0.33) points because there are three
correct answers in this question.
3. RESULTS
The research findings related to the preschool teachers’ pedagogical content knowledge on
mathematics are reported in the following figures and tables.
3.1. Results for Pre-Application
3.1.1. Reliability
Each case study in the PTPCKSM and the reliability coefficient calculation for the whole scale
are shown in Table 3. Table 3 shows that the reliability coefficient for each case study was
found to be more than .70, and that the reliability coefficient levels of the whole scale and each
case study were therefore considered “high” (Büyüköztürk, 2010). The reliability coefficients
of the case studies were identified as varying from .94 to .96; and the reliability coefficient for
the whole scale was found to be .95.
Table 3. Reliability Coefficients of PTPCKSM
Case Study Number of Items Reliability coefficient
Case Study 1 7 .95
Case Study 2 7 .94
Case Study 3 7 .96
Case Study 4 7 .94
Case Study 5 7 .96
Whole Scale 35 .95
Item difficulty and discrimination indices for the items in each case study are presented as
shown in Table 4.
Table 4. Item-level Statistics Related to PTPCKSM Case Studies
Item Case Study 1 Case Study 2 Case Study 3 Case Study 4 Case Study 5
p r p r p r p r p r
Item 1 .49 .69 .70 .57 .56 .79 .53 .87 .51 .90
Item 2 .51 .85 .45 .91 .38 .93 .50 .71 .38 .90
Item 3 .42 .95 .37 .92 .35 .94 .36 .44 .43 .92
Item 4 .38 .88 .46 .84 .40 .92 .38 .91 .57 .81
Item 5 .40 .93 .38 .86 .39 .91 .45 .93 .45 .90
Item 6 .53 .78 .39 .90 .47 .85 .44 .91 .34 .72
Item 7 .42 .89 .36 .91 .37 .78 .47 .87 .45 .88
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Table 4 shows the item difficulty index values (p) and the item discrimination index values (r)
of the items in Case Study 1. Basol (2015) classified item difficulty as “extremely easy”
(p = .85 to 1.00), “easy” (p = .61 to .84), “medium” (p = .40 to .60), “difficult” (p = .16 to .39),
and “extremely difficult” (p = .00 to .15). For Case Study 1, the item difficulties differed from
.38 to .53 and were therefore classed as either medium (p = .49, .51, .42, .40, .53, .42) or
difficult (p = .38). The item discrimination index values in Case Study 1 varied between .69
and .95.
The item difficulty index values of Case Study 2, varied between .36 and .70, with item
difficulties easy (p = .70), medium (p = .47, .40) or difficult (p = .39, .38, .37, .35). The item
discrimination index values for Case Study 2 varied between .57 and .92.
The item difficulty index values of Case Study 3 varied between .35 and .56, with item
difficulties either medium (p = .56, .46, .45) or difficult (p = .39, .38, .37, .36). The item
discrimination index values for Case Study 3 varied between .78 and .94.
The item difficulty index values of Case Study 4 varied between .36 and .53, with item
difficulties either medium (p = .53, .50, .47, .45, .44) or difficult (p = .38, .36). The item
discrimination index values for Case Study 4 varied between .44 and .93.
The item difficulty index values of Case Study 5 varied between .34 and .57, with item
difficulties either medium (p = .53, .51, .45, .43) or difficult (p = .38, .34). The item
discrimination index values for Case Study 5 varied between .72 and .90.
The results indicated that the questions were classified as either difficult, medium or easy, and
that the item discrimination index values were found to have more than .30 of variance
explained by the scale (Thorndike, 2005).
3.1.2. Validity
The content and the construct validity indices were examined through exploratory and
confirmatory factor analyses.
3.1.2.1. Content Validity
The opinion of seven experts was sought in order to assess the content validity of the
PTPCKSM. All of the items were accepted by the experts.
The content validity rate for each item was determined based on the evaluation of the expert
opinion. Afterwards, the content validity index value was determined by taking the average of
the calculated rates. The index value for each item was then used by the experts to determine
whether or not the item was deemed necessary (Büyüköztürk, 2010; Yurdugül, 2005).
The content validity index value was calculated for the eligibility level of the scale items as a
whole. With seven experts, scales with a content validity index value of more than .99 can
assure scope validity (Yurdugül, 2005). From calculation of the content validity index values
for the PTPCKSM, the eligibility level of the items in terms of their intended purpose and the
level of the children was calculated as “+1.” This value shows that all items in the PTPCKSM
were deemed to be necessary, and that the scale’s content validity was assured as a whole.
3.1.2.2. Construct Validity
Exploratory factor analysis (EFA) of the scale was conducted in order to demonstrate the
construct validity of the PTPCKSM at the pre-application stage. Both the Kaiser-Meyer-Olkin
and Bartlett tests were performed to understand whether or not the scale was appropriate for
factor analysis. For ensuring factor analysis of a scale, the Kaiser-Meyer-Olkin result should be
.50 or above, and the Bartlett Sphericity result should be statistically significant (p < .01)
(Büyüköztürk, 2010).
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627
The analysis results showed that the Kaiser-Meyer-Olkin result for the PTPCKSM was .97 and
that the Bartlett sphericity test (p < .01) was statistically significant. This result shows that
factor analysis may be performed on the scale. Upon examining the eigenvalue for both
methods, there are two factors that score as more than 1; with the first factor being 25.58 and
the second factor 1.86. These two factors were found to account for 78.41% of the total variance
in the scale, with 73.11% explained by Factor 1 and 5.30% by Factor 2. Considering the
eigenvalue and the explained variance, Factor 1 was found to be about 14 times more dominant
than Factor 2. This result therefore signified that the scale has a single factor structure. Figure 4
presents a scatter plot graph of the scale.
Figure 4. Scatter Plot Graph
Table 5. Factor Load Values as a Result of Principal Component Analysis of PTPCKSM
Item Factor Loads
Item Factor Loads
Factor 1 Factor 2 Factor 1 Factor 2
O1_3 .95 O1_7 .89
O3_3 .94 O5_7 .89
O4_5 .93 O1_4 .88
O3_2 .93 O4_7 .87
O1_5 .93 O4_1 .87
O2_3 .93 O2_5 .87
O3_4 .93 O3_6 .85
O5_3 .92 O1_2 .85
O3_5 .92 O2_4 .84
O4_4 .91 O5_4 .81
O2_2 .91 O1_6 .78
O2_7 .91 O3_1 .78
O5_2 .91 O3_7 .77
O4_6 .91 O5_6 .72
O5_1 .90 O4_2 .69 .62
O2_6 .90 O1_1 .69
O5_5 .90 O2_1 .55
O4_3 .42 .81
Number of Components
Eig
enval
ue
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When the factor loadings of each item were examined in the next stage, no item was found with
a factor load value of less than .30, and therefore no items were removed from the scale.
According to Table 5, only two factors were identified as being linked at the same time, such
as Case Study 4, Item 2 and Case Study 4, Item 3. As the difference between the Factor 1
loading (.42) and Factor 2 loading (.81) of Item O4_3 was greater than .10, it was assumed that
this problem was only due to Factor 2. As the difference between the Factor 1 loading (.69) and
Factor 2 loading (.62) of Item O4_2 (Case Study 4, Item 2) was less than .10, it was considered
that this item should be removed from the scale. However, since this item was thought to
contribute to the scale contextually (content validity), it was decided not to remove the item
from the scale.
3.2. Findings for the Main Application
As in the pre-application, the reliability coefficient was examined and the construct validity
was also tested in the main application. However, the construct validity was tested by
confirmatory factor analysis (CFA) in the main application.
3.2.1. Reliability
First, as shown in Table 6, the reliability coefficients of the scale were calculated at both the
individual case study level and for the whole scale.
Table 6. Reliability Coefficients of PTPCKSM
Case Study Number of Items Reliability coefficient
Case Study 1 7 .91
Case Study 2 7 .73
Case Study 3 7 .82
Case Study 4 7 .81
Case Study 5 7 .86
Whole Scale 35 .96
According to the findings presented in Table 6, the reliability coefficient for each of the
individual case studies was found to be more than .70, and that the reliability coefficient levels
of the scale as a whole and each case study were considered to be high (Büyüköztürk, 2010).
The reliability coefficients of the individual case studies varied from .73 to .91; and the
reliability coefficient of the whole scale was found to be .96.
3.2.2. Validity
In order to test the construct validity of the scale at the next stage, confirmatory factor analysis
(CFA) was performed using the MPlus 7.4 program, and the model produced by this analysis
is shown as Figure 5. When the goodness of fit indices of the model, it was found that the CFI
and TLI values were greater than .90, and that the RMSEA and SRMR values were less than
.08. These results showed that the model was at an acceptable level (Kline, 2016). The χ²/SD
value was calculated to be less than the accepted value of 4 (χ² (547,150) = 821.76; CFI = .91;
TLI = .91; RMSEA = .06; SRMR = .06). These results support that the scale has acceptable
construct validity (Kline, 2016). As a result, both the reliability and validity analyses results for
the main application revealed the PTPCKSM to be a suitable measurement tool.
Dağlı, Dağlıoğlu & Atalmış
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Figure 5. PTPCKSM Confirmatory Factor Analysis Model
4. DISCUSSION and CONCLUSION
Considering that mathematics is intertwined in our life skills and that the skills and processes
related to mathematics develop in children during their early years, the importance of preschool
teachers’ pedagogical knowledge about mathematics is significant. This position illustrates the
necessity for different measurement tools to assess preschool teachers’ pedagogical content
knowledge. Therefore, the current study was conducted in order to develop a new tool known
as the Preschool Teachers’ Pedagogical Content Knowledge Regarding Mathematics Scale; as
well as to perform validity and reliability studies on the developed scale.
The participants of the study were 300 preschool teachers working in formal independent
kindergartens and in nursery classes of primary/secondary schools under the Turkish Ministry
of National Education within the Kahramanmaraş Province of Turkey.
A Teacher Information Form and the Preschool Teachers’ Pedagogical Content Knowledge
Regarding Mathematics Scale, which were both developed by the researcher, were employed
as the data collection tools in this study. A pre-application study was conducted in order to
determine the clarity and responsiveness of the PTPCKSM scale items, and all of the items
were identified to have the necessary level of clarity.
For a valid scale, the problem should be well-defined, and statistically accepted values for both
validity and reliability should be assured during preparation of the scale items (Büyüköztürk,
2005). The reliability coefficients of the PTPCKSM were calculated. The findings presented in
Table 1 reveal that the reliability coefficient of each case study in the PTPCKSM to be more
than .70, and that the reliability coefficient of the whole scale was .95 which indicated the scale
to be reliable (Büyüköztürk, 2010). The item difficulty index value (p) and the item
discrimination index value (r) were calculated separately for each item in each case study of
the scale (see Table 2). When the item difficulty index values of the case studies were examined,
they were found to vary between easy, medium and difficult; whilst the item discrimination
index values were found to be higher than .30.
In addition to reliability, another requirement for determination of the scale is validity (Karasar,
2012). Therefore, explanatory and confirmatory factor analyzes indicating the construct validity
as well as the content validity analysis were performed.
Seven expert opinions were consulted to determine the content validity of PTPCKSM. All of
the items were welcomed by the experts. Content validity index was examined with the expert
opinions; as a result of the calculation of the content validity index values of PTPCKSM,
content validity index for eligibility level of the items in terms of the purpose and level of
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630
children was calculated as “+1”. This value showed that all the items in the scale were necessary
and the scale guaranteed content validity as a whole.
Kaiser-Meyer-Olkin Test result for PTPCKSM was found to be .97 and the Bartlett sphericity
test (p
Dağlı, Dağlıoğlu & Atalmış
631
ORCID
Hatice Dağlı https://orcid.org/0000-0002-0788-0413
H. Elif Dağlıoğlu https://orcid.org/0000-0002-7420-815X
E. Hasan Atalmış https://orcid.org/0000-0001-9610-491X
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